Kayla McBride scored 23 points and San Antonio overcame career-highs by Sims and Diggins to beat Tulsa 95-93 on Tuesday night.
Robinson scored 21 points, including a game-winning layup for the Stars (12-12).
"When (Robinson) gets in the gut, it's just over," Becky Hammon said. "People have to pick who they want to score at that point because somebody is going to score when she gets in there like that."
Danielle Adams added 15 points and Hammon had 10 points as San Antonio remained in third place in the West ahead of the Los Angeles Sparks (10-12).
Sims had 39 points and Diggins 34 for Tulsa (8-16), which fell to 3-9 on the road.
Sims was 15 for 20 from the field in surpassing her previous high as a rookie of 30 points against Connecticut on July 3. She fell one point short of the most ever by a rookie, set by Candace Parker in 2008.
"I just stayed aggressive," Sims said. "One thing coach was telling me was to stay aggressive from the jump, attack, come off pick-and-rolls, hit outside jumpers and just attack it. I think the lane was open a lot, so I kept taking it in there."
Diggins was 14 for 23 in besting her previous high of 33 set June 22 against Chicago.
"When people are in the zone, they are in the zone," Robinson said. "It wasn't anything scheme wise, just hoping they would miss some. They were hitting a lot of shots, a lot of tough shots over close players between two or three people. With those kinds of plays, they are just tough shots. You've just got to tip your hat."
Sims scored 12 points in the fourth quarter, including a driving layup to tie the game at 91 with 1:19 remaining. Adams followed with a pair of free throws, but Sims had another driving layup to forge another tie at 93.
"We couldn't get much stops down the stretch," Hammon said. "It was just a blow for blow kind of the game," Hammon said. "They couldn't get stops, we couldn't get stops, it was just who got the last stop."
Robinson scored the game-winner on a layup high off the glass over Glory Johnson to put the Stars ahead 95-93 with 3.5 seconds remaining. Following a Shock timeout, Sophia Malcolm-Young tipped away the inbounds pass to McBride to seal the victory, leading Tulsa coach Fred Williams to kick the scorer's table in frustration.
"I had drawn up for Skylar to X off of that and for my four (Johnson) to slip on that," Williams said. "When you see it on film, it would have been a straight layup."
Both teams had a slow start to what became an offensive showcase.
Diggins missed two open layups to start the game before making her third on a contested attempt that resulted in a three-point play. The Notre Dame product also made her team's next three shots in a personal 9-2 run that put Tulsa up 13-6 with 2:19 remaining.
Reserve post Kayla Alexander sparked San Antonio's own 9-2 run to open the second quarter, cutting Tulsa's lead to 24-21. After completing a three-point play, Alexander followed a miss by McBride to complete the run with 7:41 remaining in the half.
"Sometimes it takes a little juice," Hammon said. "I thought we were a little flat actually at the beginning. We had to get our motors running there a little bit sometimes. Getting into some people, even if it's the refs, helps."
Alexander's pull-up jumper tied the game at 30 and Robinson's 19-footer gave San Antonio its first lead at 32-30. Robinson later crossed over Sims, freeing herself for an open jumper and a 34-30 lead.
San Antonio held Tulsa to 28 percent shooting in the second quarter.
An aggressive approach helped Tulsa get nine free throws free throws in the third quarter, making all but one. The Shock tied the game at 61 on Johnson's jumper, but the Stars closed with a 3-pointer by Adams and a driving layup by Robinson for a 66-61 lead.